What is the anode net charge on a typical AAA battery?

I mean on the actually anode, what is the usual net charge around? How many extra electrons to protons are there? The reason I ask is because I'm confused what causes the voltage, is it the higher charge density relative to the other plate?

I'm not that knowledgeable about this so maybe someone will correct me soon, but I don't think there is an excess charge on the plates of a battery. I believe that the current is caused by a redox reaction. Basically chemicals at both plates are reacting with each other. The plates just facility that reaction by transporting electrons from one side to the other.

Evanish is right in that the potential is due to redox reactions, nevertheless there is an excess charge on the plates of the battery.
Specifically, there is a charge double layer on each electrodehttp://en.wikipedia.org/wiki/Double_layer_(interfacial)
On a path starting in the electrode and going deep into the electrolyte, the voltage jumps on crossing the double layer on a distance known as the Debye length (see the article) d. I would guess it to be smaller than 1 mum. http://en.wikipedia.org/wiki/Double_layer_(interfacial)
Hence if you know the voltage U of the battery and the surface A of the anode, you can get an estimation of the charge on the anode using formulas for the capacity of a condenser with these dimensions. Namely Q=UC and ##C=\epsilon A/d##, where ##\epsilon## is the dielectric constant of the solvent, e.g. of water. So with the assumed Debye length d and A being some cm^2, i would guess C to be about 1 ##\mu F##. So Q will be of the order of 1 micro Coulomb.
Without a better knowledge of the Debye length, this is only an educated guess and could be wrong by one or even two orders of magnitude.